chapter 1.4

Chapter 1.4

Quadratic Equations

Quadratic Equation in One Variable

An equation that can be written in the form

ax2 + bx + c = 0

where a, b, and c, are real numbers, is a quadratic equation

A quadratic equation is a second-degree equation—that is, an equation with a squared term and no terms of greater degree.

x2 =25, 4x2 + 4x – 5 = 0,3x2 = 4x - 8

A quadratic equation written in the form

ax2 + bx + c = 0 is in standard form.

Solving a Quadratic EquationFactoring is the simplest method of solving a quadratic equation (but one not always easily applied).

This method depends on the zero-factor property.

Zero-Factor Property

If two numbers have a product of 0 then at least one of the numbers must be zero

If ab= 0 then a = 0 or b = 0

Example 1. Using the zero factor property.

Solve 6x2 + 7x = 3

A quadratic equation of the form x2 = k can also be solved by factoring.

x2 = kx2 – k=0

0 kxkx

0 kx 0or kx

kx kx or

property.root square theproves This

Square root property

If x2 = k, then

kx kx or

Example 2 Using the Square Root Property

Solve each quadratic equation.x2 = 17

Example 2 Using the Square Root Property

Solve each quadratic equation.x2 = -25

Example 2 Using the Square Root Property

Solve each quadratic equation.(x-4)2 = 12

Completing the Square

Any quadratic equation can be solved by the method of completing the square.

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 4

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

2)2( x

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

18)2( 2 x

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

18)2( 2 x

182x

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

18)2( 2 x

182x 29

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

18)2( 2 x

182x 29 23

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

18)2( 2 x

182x 29 23

232x

Example 3 Using the Method of Completing the Square, a = 1Solve x2 – 4x – 14 = 0

144xx2 24

2

22 441444xx2 81

18)2( 2 x

182x 29 23

232x

232x

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

091x

34x2

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

91 x

34x2

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

91 x

34x2

2

34

21

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

91 x

34x2

2

34

21

2

32

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

91 x

34x2

2

34

21

2

32

94

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

91 x

34x2

2

34

21

2

32

94

94

91

94x

34x2

Example 4 Using the Method of Completing the Square, a ≠1

Solve 9x2 – 12x – 1 = 0

091x

912x2

91 x

34x2

2

34

21

2

32

94

94

91

94x

34x2

95

Example 4 Using the Method of Completing the Square, a ≠1

95

94x

34x2

Example 4 Using the Method of Completing the Square, a ≠1

95

94x

34x2

95

32-x

2

Example 4 Using the Method of Completing the Square, a ≠1

95

94x

34x2

95

32-x

2

95

32-x

Example 4 Using the Method of Completing the Square, a ≠1

95

94x

34x2

95

32-x

2

95

32-x

95

32x

Example 4 Using the Method of Completing the Square, a ≠1

95

94x

34x2

95

32-x

2

95

32-x

95

32x

95

32x

Example 4 Using the Method of Completing the Square, a ≠1

95

94x

34x2

95

32-x

2

95

32-x

95

32x

95

32x

35

32x

Example 4 Using the Method of Completing the Square, a ≠1

95

94x

34x2

95

32-x

2

95

32-x

95

32x

95

32x

35

32x

352x

The Quadratic Formula

02 cbxax

aacbbx 2

42

Watch the derivation

Example 5 Using the Quadratic Formula(Real Solutions)Solve x2 -4x = -2

Example 6 Using the Quadratic Formula(Non-real Complex Solutions)Solve 2x2 = x – 4

Example 7 Solving a Cubic EquationSolve x3 + 8 = 0

Example 8 Solving a Variable That is SquaredSolve for the specified variable.

ddA for ,4 2

Example 8 Solving a Variable That is SquaredSolve for the specified variable.

trkstrt for ),0(2

The Discriminant The quantity under the radical in the quadratic formula, b2 -4ac, is called the discriminant.

aacbbx

242

Discriminant

Then the numbers a, b, and c are integers, the value of the discriminant can be used to determine whether the solution of a quadratic equation are rational, irrational, or nonreal complex numbers, as shown in the following table.

Discriminant Number of Solutions Kind of Solutions

Positive (Perfect Square)

Positive (but not a Perfect Square)

Zero

Negative

Two

Two

One (a double solution)

Two

Rational

Irrational

Rational

Nonreal complex

Example 9 Using the DiscriminantDetermine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers. 5x2 + 2x – 4 = 0

aacbbx

242

) (2) )( (4) () ( 2

x

a

bc

Example 9 Using the DiscriminantDetermine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers. x2 – 10x = -25

aacbbx

242

) (2) )( (4) () ( 2

x

a

bc

Example 9 Using the DiscriminantDetermine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers. 2x2 – x + 1 = 0

aacbbx

242

) (2) )( (4) () ( 2

x

a

bc

Homework 1.4 # 1-79